How to model component pieces of a geometric solid?
I accomplished this. I have removed some of the facets to show the inner side.



This is how:


Make a football (as you said, subsurf an icosahedron, dissolve all vertices on middle and edges).
Add the edge split modifier with split angle 1. This will make each ngon (pentagon and hexagon) separate from its neigbours.
Apply the modifier.
Go into edit mode and select all faces.
Extrude all faces but right click directly. This will make them double but select all faces that will form the outer shell.
Scale out the selected faces.